510 
PliOFESSOR G. IL DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
3 
It is obvious then that our expression (54) is equal to — ^^2 (^o) (*^o)- Then 
3 1 -V B 
since 
OLO.' 3(1 — /3) 
, we have the surface density of the focaloid given by 
3(i + g ).pv(v-i)(v-;4|) /lY 
^ ■ 1 + i? ~ 
where 
But since 
we have 
X/(\ 
^2 (^o) ^2' (^0) 
2/0“ 
'-'-'7b 
- r-!)’ v{y;- - i)= Vvi' 
+ 
2 
•^0 
+ 
2/o' 
^ /."(v-i) 
_U ^ — 1 
- 1 ) ^ 
■ ^ 
I •-> 1 L±_? V 2 „ .■> 
y;_ - -I- I - ^' ** I _ yp' 
Z’-v p (vq- -ly 
(55) , 
(56) . 
With the object rtf writing this function in surface harmonics, and besides to 
enable us to express a rrttation potential in similar form, we have to reduce x^, y~, z" 
in the required manner. 
I now drop the suffix zero, since we are not concerned with any particular 
ellipsoid. 
Beferring again to §7, (18) and (24), we have 
(fo {(fi) = I — —^ ^ cos 2(f), 
If then we put 
2(1 - B) 
/3 ’ 
€' = 2 , 
we may write 
€. 2 {<f>) = e cos" (f) + C, 
Let us assume, if possible, 
— j;~ 
I + 
I = 
r = 
B — 1 + /3 ^ 
id ’ 
B - I - 3y8 
3/3 
(2r22((^) = y cos-(j) + r. 
k 
—rTX\ = P P2 (/^) (</>) -I" ^^2' (/^) ^2“ {^) + 
- (ftsI W — V^r) 
or (,x‘= - 7+ 'f)aos^4> = + Y){ecos-<p + £) 
From which it follows that 
Fai -f Gal' = 0, 
Fae Grx'e = 1, 
+ G [ol'ix^ y') (e' COS' <f> C) H. 
Fyl + Gyt + LT = 0, 
Fy( + Gye = — ^ 
+ /3 
- yd- 
